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In the theory of second-order, nonlinear elliptic and parabolic equations,obtaining local or global gradient bounds is often a key step for proving theexistence of solutions but it may be even more useful in many applications, forexample to singular perturbations problems. The classical Bernstein's method isa well-known tool to obtain these bounds but, in most cases, it has the defectof providing only a priori estimates. The "weak Bernstein's method", based onviscosity solutions' theory, is an alternative way to prove the globalLipschitz regularity of solutions together with some estimates but it is not soeasy to perform in the case of local bounds. The aim of this paper is toprovide an extension of the "weak Bernstein's method" which allows to provelocal gradient bounds with reasonnable technicalities.

Local Gradient Estimates for Second-Order Nonlinear Elliptic and Parabolic Equations by the Weak Bernstein's Method